Properties

Degree 2
Conductor $ 2 \cdot 7^{2} \cdot 17^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 3·5-s − 6-s + 8-s − 2·9-s − 3·10-s − 12-s − 5·13-s + 3·15-s + 16-s − 2·18-s + 4·19-s − 3·20-s + 3·23-s − 24-s + 4·25-s − 5·26-s + 5·27-s + 6·29-s + 3·30-s − 2·31-s + 32-s − 2·36-s − 4·37-s + 4·38-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s + 0.353·8-s − 2/3·9-s − 0.948·10-s − 0.288·12-s − 1.38·13-s + 0.774·15-s + 1/4·16-s − 0.471·18-s + 0.917·19-s − 0.670·20-s + 0.625·23-s − 0.204·24-s + 4/5·25-s − 0.980·26-s + 0.962·27-s + 1.11·29-s + 0.547·30-s − 0.359·31-s + 0.176·32-s − 1/3·36-s − 0.657·37-s + 0.648·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(28322\)    =    \(2 \cdot 7^{2} \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{28322} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 28322,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.8065730361\)
\(L(\frac12)\)  \(\approx\)  \(0.8065730361\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 \)
17 \( 1 \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−15.09565402766737, −14.82731506896803, −14.09890215667946, −13.79849737745488, −12.88986101819470, −12.34937256412369, −11.93826429132663, −11.75293679577890, −10.98287771762193, −10.73011420751443, −9.833565754425697, −9.323806820306987, −8.383114291824809, −7.998389013376791, −7.404759593437152, −6.784016366132843, −6.378222823378238, −5.353274176472135, −5.001386571305120, −4.609587708969239, −3.654532953950847, −3.158809489144689, −2.583204544247241, −1.427740936313802, −0.3233600560487647, 0.3233600560487647, 1.427740936313802, 2.583204544247241, 3.158809489144689, 3.654532953950847, 4.609587708969239, 5.001386571305120, 5.353274176472135, 6.378222823378238, 6.784016366132843, 7.404759593437152, 7.998389013376791, 8.383114291824809, 9.323806820306987, 9.833565754425697, 10.73011420751443, 10.98287771762193, 11.75293679577890, 11.93826429132663, 12.34937256412369, 12.88986101819470, 13.79849737745488, 14.09890215667946, 14.82731506896803, 15.09565402766737

Graph of the $Z$-function along the critical line